Chapter 35
Solutions for tenth week’s assignments

35.1 Exercises 6.4

Problem 15

The sum is 0.\overline{8}, which is rational. So the sum of two irrationals may be rational. That should not be surprising; 2 -\sqrt{2} and \sqrt{2} are irrational, but their sum is the rational 2.

Problem 16

The sum is 0.262662666\mathop{\mathop{…}}, which is not repeating or terminating and thus is irrational. So the sum of two irrationals, like \sqrt{2} and \sqrt{2}, can be irrational, like 2\sqrt{2}.

Problem 25

\sqrt{50} = \sqrt{2 ⋅ {5}^{2}} = {(2 ⋅ {5}^{2})}^{{1\over 2} } = 5\sqrt{2} ≈ 7.07.

Problem 26

\sqrt{32} = \sqrt{{2}^{5}} = {2}^{2}\sqrt{2} ≈ 5.66

Problem 27

\sqrt{75} = \sqrt{3 ⋅ {5}^{2}} = 5\sqrt{3} ≈ 8.66

Problem 28

\sqrt{150} = \sqrt{2 ⋅ 3 ⋅ {5}^{2}} = 5\sqrt{6} ≈ 12.25.

Problem 41

3\sqrt{18} + \sqrt{2} = 3 ⋅ {(2 ⋅ {3}^{2})}^{{1\over 2} } + {2}^{{1\over 2} } = 9 ⋅ {2}^{{1\over 2} } + {2}^{{1\over 2} } = 10 ⋅ {2}^{{1\over 2} } = 10\sqrt{2}

Problem 42

2\sqrt{48} -\sqrt{3} = 2 ⋅ {({2}^{4} ⋅ 3)}^{{1\over 2} } - {3}^{{1\over 2} } = {2}^{3} ⋅ {3}^{{1\over 2} } - {3}^{{1\over 2} } = 7\sqrt{3}

Problem 43

-\sqrt{12} + \sqrt{75} = -{({2}^{2} ⋅ 3)}^{{1\over 2} } + {(3 ⋅ {5}^{2})}^{{1\over 2} } = -2 ⋅ {3}^{{1\over 2} } + 5 ⋅ {3}^{{1\over 2} } = (5 - 2)\sqrt{3} = 3\sqrt{3}

Problem 44

2\sqrt{27} -\sqrt{300} = 2 ⋅ {({3}^{3})}^{{1\over 2} } - {(3 ⋅ 1{0}^{2})}^{{1\over 2} } = 6 ⋅ {3}^{{1\over 2} } - 10 ⋅ {3}^{{1\over 2} } = -4\sqrt{3}

Problem 49

P = 2π\sqrt{{5.1\over 32}} ≈ 2.5. Note that 32 is an approximation to gravitational acceleration.

Problem 81

Approximating:

1.{1}^{10}	1.0{1}^{100}	1.00{1}^{1000}	1.000{1}^{10000}	1.0000{1}^{100000}
2.5937	2.7048	2.7169	2.7181	2.7183
0.95418e	0.99505e	0.99950e	0.99095e	1.00000e

The constant e ={\mathop{ lim}}_{n→∞}{(1 + {1\over n})}^{n}, and the experimental results above bear this out. The first few digits converge very quickly.

35.2 Exercises 6.3

Problem 75

0.75

Problem 76

0.875

Problem 79

0.\overline{27}

Problem 80

0.\overline{81}

Problem 86

0.105 = {105\over 1000} = {21\over 200}

Problem 87

0.934 = {934\over 1000} = {467\over 500}

Problem 88

0.7984 = {7984\over 10000} = {499\over 625}

Problem 95

• {1\over 3} = 0.\overline{3}
• {2\over 3} = 0.\overline{6}
• {1\over 3} + {2\over 3} = 0.\overline{3} + 0.\overline{6} = 0.\overline{9}
• As covered in class, repeating nines are equal to a one one digit over, so 0.\overline{9} = 1.

Problem 96

Here, 3 ⋅ 0.\overline{3} = 0.\overline{9} = 1.

35.3 Exercises 6.5

Problem 1

3.00 ⋅ 12 = 36, true

Problem 2

0.25 = {25\over 100} = {1\over 4}, true

Problem 3

Rounding 759.367 to the second place beyond the decimal should give either 759.37 or 759.36 depending on the rounding rule. false

Problem 4

With round-to-nearest (even or upwards), this is true.

Problem 5

0.50 = {50\over 100} = {1\over 2}, true

35.4 Rounding and floating-point

35.4.1 Rounding

35.4.2 Errors in computations

Using Octave on a 64-bit Intel-based machine with the “short” display format:

• - 1.1102 ⋅ 1{0}^{-16}
• - 1.1102
• 0 : Sometimes errors cancel themselves out. Not every computational error is bad.
• 0
• - 4.4409 ⋅ 1{0}^{-16}
• - 4.4409

35.4.3 Extra digits

Using Octave on a 64-bit Intel-based machine with the “short” display format:

• 1 / 3 produces 0.33333. Then (1/3) - 0.33333 produces 3.3333e-06 or 3.3333 ⋅ 1{0}^{-6}.
• pi produces 3.1416, and pi - 3.1416 produces -7.3464e-06 or - 7.3464 ⋅ 1{0}^{-6}.